In linear regression with functional predictors and scalar responses, it may be advantageous, particularly if the function is thought to contain features at many scales, to restrict the coefficient function to the span of a wavelet basis, thereby converting the problem into one of variable selection. If the coefficient function is sparsely represented in the wavelet domain, we may employ the well-known LASSO to select a relatively small number of nonzero wavelet coefficients. This is a natural approach to take but to date, the properties of such an estimator have not been studied. In this paper we describe the wavelet-based LASSO approach to regressing scalars on functions and investigate both its asymptotic convergence and its finite-sample performance through both simulation and real-data application. We compare the performance of this approach with existing methods and find that the wavelet-based LASSO performs relatively well, particularly when the true coefficient function is spiky. Source code to implement the method and data sets used in the study are provided as supplemental materials available online.
Keywords: functional data analysis; penalized linear regression; variable selection; wavelet regression.